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arXiv:2208.03910 · Physical review. E

Semi-Markov processes in open quantum systems: Connections and applications in counting statistics

Fei Liu

Published Nov 23, 2022• 7 citations
arXivHTMLPDF

Abstract

Using the age-structure formalism, we definitely establish connections between semi-Markov processes and the dynamics of open quantum systems that satisfy the Markov quantum master equations. A generalized Feynman-Kac formula of the semi-Markov processes is also proposed. In addition to inheriting all statistical properties possessed by the piecewise deterministic processes of wave functions, the semi-Markov processes show their unique advantages in quantum counting statistics. Compared with the conventional method of the tilted quantum master equation, they can be applied to more general counting quantities. In particular, the terms involved in the method have precise probability meanings. We use a driven two-level quantum system to exemplify these results.

DOI
10.1103/physreve.106.054152

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Using the age-structure formalism, we definitely establish connections between semi-Markov processes and the dynamics of open quantum systems that satisfy the Markov quantum master equations. A generalized Feynman-Kac formula of the semi-Markov processes is also proposed. In addition to inheriting all statistical properties possessed by the piecewise deterministic processes of wave functions, the semi-Markov processes show their unique advantages in quantum counting statistics. Compared with the conventional method of the tilted quantum master equation, they can be applied to more general counting quantities. In particular, the terms involved in the method have precise probability meanings. We use a driven two-level quantum system to exemplify these results.
A semi-Markov process method for obtaining general counting statistics for open quantum systems is extended to the scenario of resetting. The simultaneous presence of random resets and wave function collapses means that the quantum jump trajectories are no longer semi-Markov. However, focusing on trajectories and using simple probability formulas, general counting statistics can still be constructed from reset-free statistics. An exact tilted matrix equation is also obtained. The inputs of these methods are the survival distributions and waiting-time density distributions instead of quantum operators. In addition, a continuous-time cloning algorithm is introduced to simulate the large-deviation properties of open quantum systems. Several quantum optics systems are used to demonstrate these results.
Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of quantum evolutions, which is a direct generalisation of the corresponding classical concept, guarantees mathematically well-defined master equations, while accounting for a wide range of phenomena, possibly in the non-Markovian regime. In particular, we analyse the emergence of a dephasing term when moving from one type of master equation to the other, by means of several examples. We also investigate the corresponding Redfield-like approximated dynamics, which are obtained after a coarse graining in time. Relying on general properties of the associated classical random process, we conclude that such an approximation always leads to a Markovian evolution for the considered class of dynamics.
In a specific class of open quantum systems with finite and fixed numbers of collapsed quantum states, the semi-Markov process method is used to calculate the large deviations of the first passage time statistics. The core formula is an equation of poles, which is also applied in determining the scaled generating functions (SCGFs) of the counting statistics. For simple counting variables, the SCGFs of the first passage time statistics are derived by finding the largest modulus of the roots of this equation with respect to the z-transform parameter and then calculating its logarithm. The procedure is analogous to that of solving for the SCGFs of the counting statistics. However, for current-like variables, the method generally fails unless the equation of pole is simplified to a quadratic form. The fundamental reason for this lies in the nonuniqueness between the roots and the region of convergence for the joint transform. We illustrate these results via a resonantly driven two-level quantum system, where for several counting variables the solutions to the SCGFs of the first passage time are analytically obtained. Furthermore, we apply these functions to investigate quantum violations of the classical kinetic and thermodynamic uncertainty relations.
We construct a large class of non-Markovian master equations that describe the dynamics of open quantum systems featuring strong memory effects, which relies on a quantum generalization of the concept of classical semi-Markov processes. General conditions for the complete positivity of the corresponding quantum dynamical maps are formulated. The resulting non-Markovian quantum processes allow the treatment of a variety of physical systems, as is illustrated by means of various examples and applications, including quantum optical systems and models of quantum transport.
Semi-Markov processes represent a well-known and widely used class of random processes in classical probability theory. Here, we develop an extension of this type of non-Markovian dynamics to the quantum regime. This extension is demonstrated to yield quantum master equations with memory kernels which allow the formulation of explicit conditions for the complete positivity of the corresponding quantum dynamical maps, thus leading to important insights into the structural characterization of the non-Markovian quantum dynamics of open systems. Explicit examples are analyzed in detail.
We provide a large class of quantum evolutions governed by the memory kernel master equation. This class defines a quantum analog of so-called semi-Markov classical stochastic dynamics. In this paper we provide a precise definition of quantum semi-Markov evolution, and using the appropriate gauge freedom we propose a suitable generalization which contains a majority of examples considered so far in the literature. The key concepts are quantum counterparts of classical waiting time distribution and survival probability---a quantum waiting time operator and a quantum survival operator, respectively. In particular collision models and their generalizations considered recently are special examples of generalized semi-Markov evolution. This approach allows for an interesting generalization of the trajectory description of the quantum dynamics in terms of positive operator-valued measure densities.